The ratio of the present ages of Rahul and his sister is 3 : 4. Ten years ago, the ratio of their ages was 13 : 19. What is the present age, in years, of Rahul?

Introduction / Context:
This question involves comparing age ratios at the present time and in the past. It is a typical problem in the topic of ages where simple algebra and careful interpretation of ratios are required. By converting each ratio statement into an equation involving a common variable and then using the time shift of ten years, we can determine the actual present age of Rahul.

Given Data / Assumptions:

• The present age ratio of Rahul and his sister is 3 : 4.
• Ten years ago, the ratio of their ages was 13 : 19.
• We are asked to find the present age of Rahul in years.
• All ages are positive and measured in years.

Concept / Approach:
When ages are given in ratio form, we model them as multiples of a common factor. Since there is a time difference of ten years, we subtract ten from each present age expression to get past ages. We then set the fraction of past ages equal to the given past ratio. This yields a single equation in one variable, which can be solved to get the common multiple and thus the actual age of Rahul.

Step-by-Step Solution:
Step 1: Let the present age of Rahul be 3x years and the present age of his sister be 4x years. Step 2: Ten years ago, Rahul age would have been 3x - 10 years. Step 3: Ten years ago, his sister age would have been 4x - 10 years. Step 4: At that time, the ratio of their ages is given as 13 : 19, so (3x - 10) / (4x - 10) = 13 / 19. Step 5: Cross multiply to get 19 * (3x - 10) = 13 * (4x - 10). Step 6: Expand both sides: 57x - 190 = 52x - 130. Step 7: Rearrange terms: 57x - 52x = -130 + 190, so 5x = 60. Step 8: Divide by 5 to get x = 12. Step 9: Therefore, Rahul present age is 3x = 3 * 12 = 36 years.

Verification / Alternative check:
Check the ages ten years ago. Rahul would be 36 - 10 = 26 years, and his sister would be 4 * 12 - 10 = 48 - 10 = 38 years. The ratio 26 : 38 can be simplified by dividing both numbers by 2 to get 13 : 19, which exactly matches the given past ratio. This confirms that our calculation is correct and that Rahul present age is indeed 36 years.

Why Other Options Are Wrong:
Option B (48 years) would produce a different past ratio that does not simplify to 13 : 19 when paired with the corresponding age of his sister.
Option C (42 years) similarly fails to produce the required past ratio and therefore is incorrect.
Option D (54 years) leads to past ages that do not satisfy the ratio 13 : 19, making this option invalid as well.

Common Pitfalls:
One common mistake is to add or subtract the time difference from only one of the ages, but both individuals age equally over time. Another pitfall is to treat ratios as simple differences instead of fractional relationships, which causes errors in forming the equations. Always write the ratio as a fraction and then cross multiply carefully to avoid algebraic mistakes.

Final Answer:
The present age of Rahul is 36 years.